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1 |
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2 theory Positions |
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3 imports "Spec" "Lexer" |
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4 begin |
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5 |
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6 chapter \<open>An alternative definition for POSIX values\<close> |
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7 |
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8 section \<open>Positions in Values\<close> |
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9 |
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10 fun |
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11 at :: "val \<Rightarrow> nat list \<Rightarrow> val" |
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12 where |
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13 "at v [] = v" |
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14 | "at (Left v) (0#ps)= at v ps" |
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15 | "at (Right v) (Suc 0#ps)= at v ps" |
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16 | "at (Seq v1 v2) (0#ps)= at v1 ps" |
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17 | "at (Seq v1 v2) (Suc 0#ps)= at v2 ps" |
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18 | "at (Stars vs) (n#ps)= at (nth vs n) ps" |
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19 |
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20 |
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21 |
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22 fun Pos :: "val \<Rightarrow> (nat list) set" |
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23 where |
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24 "Pos (Void) = {[]}" |
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25 | "Pos (Char c) = {[]}" |
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26 | "Pos (Left v) = {[]} \<union> {0#ps | ps. ps \<in> Pos v}" |
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27 | "Pos (Right v) = {[]} \<union> {1#ps | ps. ps \<in> Pos v}" |
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28 | "Pos (Seq v1 v2) = {[]} \<union> {0#ps | ps. ps \<in> Pos v1} \<union> {1#ps | ps. ps \<in> Pos v2}" |
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29 | "Pos (Stars []) = {[]}" |
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30 | "Pos (Stars (v#vs)) = {[]} \<union> {0#ps | ps. ps \<in> Pos v} \<union> {Suc n#ps | n ps. n#ps \<in> Pos (Stars vs)}" |
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31 |
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32 |
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33 lemma Pos_stars: |
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34 "Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})" |
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35 apply(induct vs) |
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36 apply(auto simp add: insert_ident less_Suc_eq_0_disj) |
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37 done |
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38 |
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39 lemma Pos_empty: |
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40 shows "[] \<in> Pos v" |
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41 by (induct v rule: Pos.induct)(auto) |
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42 |
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43 |
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44 abbreviation |
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45 "intlen vs \<equiv> int (length vs)" |
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46 |
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47 |
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48 definition pflat_len :: "val \<Rightarrow> nat list => int" |
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49 where |
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50 "pflat_len v p \<equiv> (if p \<in> Pos v then intlen (flat (at v p)) else -1)" |
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51 |
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52 lemma pflat_len_simps: |
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53 shows "pflat_len (Seq v1 v2) (0#p) = pflat_len v1 p" |
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54 and "pflat_len (Seq v1 v2) (Suc 0#p) = pflat_len v2 p" |
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55 and "pflat_len (Left v) (0#p) = pflat_len v p" |
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56 and "pflat_len (Left v) (Suc 0#p) = -1" |
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57 and "pflat_len (Right v) (Suc 0#p) = pflat_len v p" |
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58 and "pflat_len (Right v) (0#p) = -1" |
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59 and "pflat_len (Stars (v#vs)) (Suc n#p) = pflat_len (Stars vs) (n#p)" |
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60 and "pflat_len (Stars (v#vs)) (0#p) = pflat_len v p" |
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61 and "pflat_len v [] = intlen (flat v)" |
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62 by (auto simp add: pflat_len_def Pos_empty) |
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63 |
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64 lemma pflat_len_Stars_simps: |
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65 assumes "n < length vs" |
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66 shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p" |
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67 using assms |
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68 apply(induct vs arbitrary: n p) |
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69 apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps) |
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70 done |
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71 |
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72 lemma pflat_len_outside: |
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73 assumes "p \<notin> Pos v1" |
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74 shows "pflat_len v1 p = -1 " |
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75 using assms by (simp add: pflat_len_def) |
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76 |
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77 |
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78 |
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79 section \<open>Orderings\<close> |
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80 |
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81 |
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82 definition prefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubseteq>pre _" [60,59] 60) |
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83 where |
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84 "ps1 \<sqsubseteq>pre ps2 \<equiv> \<exists>ps'. ps1 @ps' = ps2" |
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85 |
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86 definition sprefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubset>spre _" [60,59] 60) |
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87 where |
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88 "ps1 \<sqsubset>spre ps2 \<equiv> ps1 \<sqsubseteq>pre ps2 \<and> ps1 \<noteq> ps2" |
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89 |
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90 inductive lex_list :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" ("_ \<sqsubset>lex _" [60,59] 60) |
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91 where |
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92 "[] \<sqsubset>lex (p#ps)" |
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93 | "ps1 \<sqsubset>lex ps2 \<Longrightarrow> (p#ps1) \<sqsubset>lex (p#ps2)" |
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94 | "p1 < p2 \<Longrightarrow> (p1#ps1) \<sqsubset>lex (p2#ps2)" |
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95 |
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96 lemma lex_irrfl: |
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97 fixes ps1 ps2 :: "nat list" |
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98 assumes "ps1 \<sqsubset>lex ps2" |
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99 shows "ps1 \<noteq> ps2" |
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100 using assms |
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101 by(induct rule: lex_list.induct)(auto) |
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102 |
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103 lemma lex_simps [simp]: |
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104 fixes xs ys :: "nat list" |
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105 shows "[] \<sqsubset>lex ys \<longleftrightarrow> ys \<noteq> []" |
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106 and "xs \<sqsubset>lex [] \<longleftrightarrow> False" |
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107 and "(x # xs) \<sqsubset>lex (y # ys) \<longleftrightarrow> (x < y \<or> (x = y \<and> xs \<sqsubset>lex ys))" |
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108 by (auto simp add: neq_Nil_conv elim: lex_list.cases intro: lex_list.intros) |
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109 |
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110 lemma lex_trans: |
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111 fixes ps1 ps2 ps3 :: "nat list" |
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112 assumes "ps1 \<sqsubset>lex ps2" "ps2 \<sqsubset>lex ps3" |
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113 shows "ps1 \<sqsubset>lex ps3" |
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114 using assms |
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115 by (induct arbitrary: ps3 rule: lex_list.induct) |
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116 (auto elim: lex_list.cases) |
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117 |
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118 |
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119 lemma lex_trichotomous: |
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120 fixes p q :: "nat list" |
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121 shows "p = q \<or> p \<sqsubset>lex q \<or> q \<sqsubset>lex p" |
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122 apply(induct p arbitrary: q) |
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123 apply(auto elim: lex_list.cases) |
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124 apply(case_tac q) |
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125 apply(auto) |
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126 done |
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127 |
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128 |
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129 |
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130 |
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131 section \<open>POSIX Ordering of Values According to Okui \& Suzuki\<close> |
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132 |
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133 |
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134 definition PosOrd:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val _ _" [60, 60, 59] 60) |
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135 where |
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136 "v1 \<sqsubset>val p v2 \<equiv> pflat_len v1 p > pflat_len v2 p \<and> |
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137 (\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)" |
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138 |
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139 lemma PosOrd_def2: |
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140 shows "v1 \<sqsubset>val p v2 \<longleftrightarrow> |
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141 pflat_len v1 p > pflat_len v2 p \<and> |
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142 (\<forall>q \<in> Pos v1. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q) \<and> |
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143 (\<forall>q \<in> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)" |
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144 unfolding PosOrd_def |
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145 apply(auto) |
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146 done |
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147 |
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148 |
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149 definition PosOrd_ex:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubset>val _" [60, 59] 60) |
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150 where |
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151 "v1 :\<sqsubset>val v2 \<equiv> \<exists>p. v1 \<sqsubset>val p v2" |
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152 |
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153 definition PosOrd_ex_eq:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubseteq>val _" [60, 59] 60) |
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154 where |
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155 "v1 :\<sqsubseteq>val v2 \<equiv> v1 :\<sqsubset>val v2 \<or> v1 = v2" |
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156 |
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157 |
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158 lemma PosOrd_trans: |
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159 assumes "v1 :\<sqsubset>val v2" "v2 :\<sqsubset>val v3" |
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160 shows "v1 :\<sqsubset>val v3" |
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161 proof - |
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162 from assms obtain p p' |
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163 where as: "v1 \<sqsubset>val p v2" "v2 \<sqsubset>val p' v3" unfolding PosOrd_ex_def by blast |
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164 then have pos: "p \<in> Pos v1" "p' \<in> Pos v2" unfolding PosOrd_def pflat_len_def |
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165 by (smt not_int_zless_negative)+ |
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166 have "p = p' \<or> p \<sqsubset>lex p' \<or> p' \<sqsubset>lex p" |
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167 by (rule lex_trichotomous) |
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168 moreover |
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169 { assume "p = p'" |
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170 with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def |
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171 by (smt Un_iff) |
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172 then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast |
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173 } |
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174 moreover |
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175 { assume "p \<sqsubset>lex p'" |
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176 with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def |
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177 by (smt Un_iff lex_trans) |
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178 then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast |
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179 } |
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180 moreover |
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181 { assume "p' \<sqsubset>lex p" |
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182 with as have "v1 \<sqsubset>val p' v3" unfolding PosOrd_def |
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183 by (smt Un_iff lex_trans pflat_len_def) |
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184 then have "v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast |
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185 } |
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186 ultimately show "v1 :\<sqsubset>val v3" by blast |
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187 qed |
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188 |
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189 lemma PosOrd_irrefl: |
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190 assumes "v :\<sqsubset>val v" |
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191 shows "False" |
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192 using assms unfolding PosOrd_ex_def PosOrd_def |
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193 by auto |
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194 |
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195 lemma PosOrd_assym: |
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196 assumes "v1 :\<sqsubset>val v2" |
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197 shows "\<not>(v2 :\<sqsubset>val v1)" |
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198 using assms |
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199 using PosOrd_irrefl PosOrd_trans by blast |
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200 |
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201 (* |
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202 :\<sqsubseteq>val and :\<sqsubset>val are partial orders. |
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203 *) |
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204 |
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205 lemma PosOrd_ordering: |
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206 shows "ordering (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)" |
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207 unfolding ordering_def PosOrd_ex_eq_def |
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208 apply(auto) |
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209 using PosOrd_irrefl apply blast |
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210 using PosOrd_assym apply blast |
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211 using PosOrd_trans by blast |
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212 |
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213 lemma PosOrd_order: |
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214 shows "class.order (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)" |
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215 using PosOrd_ordering |
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216 apply(simp add: class.order_def class.preorder_def class.order_axioms_def) |
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217 unfolding ordering_def |
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218 by blast |
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219 |
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220 |
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221 lemma PosOrd_ex_eq2: |
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222 shows "v1 :\<sqsubset>val v2 \<longleftrightarrow> (v1 :\<sqsubseteq>val v2 \<and> v1 \<noteq> v2)" |
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223 using PosOrd_ordering |
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224 unfolding ordering_def |
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225 by auto |
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226 |
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227 lemma PosOrdeq_trans: |
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228 assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v3" |
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229 shows "v1 :\<sqsubseteq>val v3" |
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230 using assms PosOrd_ordering |
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231 unfolding ordering_def |
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232 by blast |
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233 |
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234 lemma PosOrdeq_antisym: |
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235 assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v1" |
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236 shows "v1 = v2" |
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237 using assms PosOrd_ordering |
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238 unfolding ordering_def |
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239 by blast |
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240 |
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241 lemma PosOrdeq_refl: |
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242 shows "v :\<sqsubseteq>val v" |
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243 unfolding PosOrd_ex_eq_def |
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244 by auto |
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245 |
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246 |
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247 lemma PosOrd_shorterE: |
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248 assumes "v1 :\<sqsubset>val v2" |
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249 shows "length (flat v2) \<le> length (flat v1)" |
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250 using assms unfolding PosOrd_ex_def PosOrd_def |
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251 apply(auto) |
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252 apply(case_tac p) |
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253 apply(simp add: pflat_len_simps) |
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254 apply(drule_tac x="[]" in bspec) |
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255 apply(simp add: Pos_empty) |
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256 apply(simp add: pflat_len_simps) |
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257 done |
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258 |
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259 lemma PosOrd_shorterI: |
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260 assumes "length (flat v2) < length (flat v1)" |
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261 shows "v1 :\<sqsubset>val v2" |
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262 unfolding PosOrd_ex_def PosOrd_def pflat_len_def |
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263 using assms Pos_empty by force |
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264 |
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265 lemma PosOrd_spreI: |
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266 assumes "flat v' \<sqsubset>spre flat v" |
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267 shows "v :\<sqsubset>val v'" |
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268 using assms |
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269 apply(rule_tac PosOrd_shorterI) |
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270 unfolding prefix_list_def sprefix_list_def |
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271 by (metis append_Nil2 append_eq_conv_conj drop_all le_less_linear) |
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272 |
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273 lemma pflat_len_inside: |
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274 assumes "pflat_len v2 p < pflat_len v1 p" |
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275 shows "p \<in> Pos v1" |
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276 using assms |
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277 unfolding pflat_len_def |
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278 by (auto split: if_splits) |
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279 |
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280 |
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281 lemma PosOrd_Left_Right: |
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282 assumes "flat v1 = flat v2" |
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283 shows "Left v1 :\<sqsubset>val Right v2" |
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284 unfolding PosOrd_ex_def |
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285 apply(rule_tac x="[0]" in exI) |
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286 apply(auto simp add: PosOrd_def pflat_len_simps assms) |
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287 done |
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288 |
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289 lemma PosOrd_LeftE: |
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290 assumes "Left v1 :\<sqsubset>val Left v2" "flat v1 = flat v2" |
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291 shows "v1 :\<sqsubset>val v2" |
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292 using assms |
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293 unfolding PosOrd_ex_def PosOrd_def2 |
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294 apply(auto simp add: pflat_len_simps) |
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295 apply(frule pflat_len_inside) |
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296 apply(auto simp add: pflat_len_simps) |
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297 by (metis lex_simps(3) pflat_len_simps(3)) |
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298 |
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299 lemma PosOrd_LeftI: |
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300 assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2" |
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301 shows "Left v1 :\<sqsubset>val Left v2" |
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302 using assms |
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303 unfolding PosOrd_ex_def PosOrd_def2 |
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304 apply(auto simp add: pflat_len_simps) |
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305 by (metis less_numeral_extra(3) lex_simps(3) pflat_len_simps(3)) |
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306 |
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307 lemma PosOrd_Left_eq: |
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308 assumes "flat v1 = flat v2" |
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309 shows "Left v1 :\<sqsubset>val Left v2 \<longleftrightarrow> v1 :\<sqsubset>val v2" |
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310 using assms PosOrd_LeftE PosOrd_LeftI |
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311 by blast |
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312 |
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313 |
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314 lemma PosOrd_RightE: |
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315 assumes "Right v1 :\<sqsubset>val Right v2" "flat v1 = flat v2" |
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316 shows "v1 :\<sqsubset>val v2" |
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317 using assms |
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318 unfolding PosOrd_ex_def PosOrd_def2 |
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319 apply(auto simp add: pflat_len_simps) |
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320 apply(frule pflat_len_inside) |
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321 apply(auto simp add: pflat_len_simps) |
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322 by (metis lex_simps(3) pflat_len_simps(5)) |
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323 |
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324 lemma PosOrd_RightI: |
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325 assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2" |
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326 shows "Right v1 :\<sqsubset>val Right v2" |
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327 using assms |
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328 unfolding PosOrd_ex_def PosOrd_def2 |
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329 apply(auto simp add: pflat_len_simps) |
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330 by (metis lex_simps(3) nat_neq_iff pflat_len_simps(5)) |
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331 |
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332 |
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333 lemma PosOrd_Right_eq: |
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334 assumes "flat v1 = flat v2" |
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335 shows "Right v1 :\<sqsubset>val Right v2 \<longleftrightarrow> v1 :\<sqsubset>val v2" |
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336 using assms PosOrd_RightE PosOrd_RightI |
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337 by blast |
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338 |
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339 |
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340 lemma PosOrd_SeqI1: |
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341 assumes "v1 :\<sqsubset>val w1" "flat (Seq v1 v2) = flat (Seq w1 w2)" |
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342 shows "Seq v1 v2 :\<sqsubset>val Seq w1 w2" |
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343 using assms(1) |
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344 apply(subst (asm) PosOrd_ex_def) |
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345 apply(subst (asm) PosOrd_def) |
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346 apply(clarify) |
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347 apply(subst PosOrd_ex_def) |
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348 apply(rule_tac x="0#p" in exI) |
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349 apply(subst PosOrd_def) |
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350 apply(rule conjI) |
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351 apply(simp add: pflat_len_simps) |
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352 apply(rule ballI) |
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353 apply(rule impI) |
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354 apply(simp only: Pos.simps) |
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355 apply(auto)[1] |
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356 apply(simp add: pflat_len_simps) |
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357 apply(auto simp add: pflat_len_simps) |
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358 using assms(2) |
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359 apply(simp) |
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360 apply(metis length_append of_nat_add) |
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361 done |
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362 |
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363 lemma PosOrd_SeqI2: |
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364 assumes "v2 :\<sqsubset>val w2" "flat v2 = flat w2" |
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365 shows "Seq v v2 :\<sqsubset>val Seq v w2" |
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366 using assms(1) |
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367 apply(subst (asm) PosOrd_ex_def) |
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368 apply(subst (asm) PosOrd_def) |
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369 apply(clarify) |
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370 apply(subst PosOrd_ex_def) |
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371 apply(rule_tac x="Suc 0#p" in exI) |
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372 apply(subst PosOrd_def) |
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373 apply(rule conjI) |
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374 apply(simp add: pflat_len_simps) |
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375 apply(rule ballI) |
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376 apply(rule impI) |
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377 apply(simp only: Pos.simps) |
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378 apply(auto)[1] |
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379 apply(simp add: pflat_len_simps) |
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380 using assms(2) |
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381 apply(simp) |
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382 apply(auto simp add: pflat_len_simps) |
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383 done |
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384 |
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385 lemma PosOrd_Seq_eq: |
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386 assumes "flat v2 = flat w2" |
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387 shows "(Seq v v2) :\<sqsubset>val (Seq v w2) \<longleftrightarrow> v2 :\<sqsubset>val w2" |
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388 using assms |
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389 apply(auto) |
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390 prefer 2 |
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391 apply(simp add: PosOrd_SeqI2) |
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392 apply(simp add: PosOrd_ex_def) |
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393 apply(auto) |
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394 apply(case_tac p) |
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395 apply(simp add: PosOrd_def pflat_len_simps) |
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396 apply(case_tac a) |
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397 apply(simp add: PosOrd_def pflat_len_simps) |
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398 apply(clarify) |
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399 apply(case_tac nat) |
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400 prefer 2 |
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401 apply(simp add: PosOrd_def pflat_len_simps pflat_len_outside) |
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402 apply(rule_tac x="list" in exI) |
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403 apply(auto simp add: PosOrd_def2 pflat_len_simps) |
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404 apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2)) |
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405 apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2)) |
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406 done |
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407 |
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408 |
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409 |
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410 lemma PosOrd_StarsI: |
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411 assumes "v1 :\<sqsubset>val v2" "flats (v1#vs1) = flats (v2#vs2)" |
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412 shows "Stars (v1#vs1) :\<sqsubset>val Stars (v2#vs2)" |
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413 using assms(1) |
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414 apply(subst (asm) PosOrd_ex_def) |
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415 apply(subst (asm) PosOrd_def) |
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416 apply(clarify) |
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417 apply(subst PosOrd_ex_def) |
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418 apply(subst PosOrd_def) |
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419 apply(rule_tac x="0#p" in exI) |
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420 apply(simp add: pflat_len_Stars_simps pflat_len_simps) |
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421 using assms(2) |
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422 apply(simp add: pflat_len_simps) |
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423 apply(auto simp add: pflat_len_Stars_simps pflat_len_simps) |
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424 by (metis length_append of_nat_add) |
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425 |
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426 lemma PosOrd_StarsI2: |
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427 assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flats vs1 = flats vs2" |
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428 shows "Stars (v#vs1) :\<sqsubset>val Stars (v#vs2)" |
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429 using assms(1) |
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430 apply(subst (asm) PosOrd_ex_def) |
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431 apply(subst (asm) PosOrd_def) |
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432 apply(clarify) |
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433 apply(subst PosOrd_ex_def) |
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434 apply(subst PosOrd_def) |
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435 apply(case_tac p) |
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436 apply(simp add: pflat_len_simps) |
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437 apply(rule_tac x="Suc a#list" in exI) |
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438 apply(auto simp add: pflat_len_Stars_simps pflat_len_simps assms(2)) |
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439 done |
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440 |
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441 lemma PosOrd_Stars_appendI: |
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442 assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)" |
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443 shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)" |
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444 using assms |
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445 apply(induct vs) |
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446 apply(simp) |
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447 apply(simp add: PosOrd_StarsI2) |
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448 done |
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449 |
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450 lemma PosOrd_StarsE2: |
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451 assumes "Stars (v # vs1) :\<sqsubset>val Stars (v # vs2)" |
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452 shows "Stars vs1 :\<sqsubset>val Stars vs2" |
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453 using assms |
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454 apply(subst (asm) PosOrd_ex_def) |
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455 apply(erule exE) |
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456 apply(case_tac p) |
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457 apply(simp) |
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458 apply(simp add: PosOrd_def pflat_len_simps) |
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459 apply(subst PosOrd_ex_def) |
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460 apply(rule_tac x="[]" in exI) |
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461 apply(simp add: PosOrd_def pflat_len_simps Pos_empty) |
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462 apply(simp) |
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463 apply(case_tac a) |
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464 apply(clarify) |
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465 apply(auto simp add: pflat_len_simps PosOrd_def pflat_len_def split: if_splits)[1] |
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466 apply(clarify) |
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467 apply(simp add: PosOrd_ex_def) |
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468 apply(rule_tac x="nat#list" in exI) |
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469 apply(auto simp add: PosOrd_def pflat_len_simps)[1] |
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470 apply(case_tac q) |
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471 apply(simp add: PosOrd_def pflat_len_simps) |
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472 apply(clarify) |
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473 apply(drule_tac x="Suc a # lista" in bspec) |
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474 apply(simp) |
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475 apply(auto simp add: PosOrd_def pflat_len_simps)[1] |
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476 apply(case_tac q) |
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477 apply(simp add: PosOrd_def pflat_len_simps) |
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478 apply(clarify) |
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479 apply(drule_tac x="Suc a # lista" in bspec) |
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480 apply(simp) |
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481 apply(auto simp add: PosOrd_def pflat_len_simps)[1] |
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482 done |
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483 |
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484 lemma PosOrd_Stars_appendE: |
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485 assumes "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)" |
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486 shows "Stars vs1 :\<sqsubset>val Stars vs2" |
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487 using assms |
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488 apply(induct vs) |
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489 apply(simp) |
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490 apply(simp add: PosOrd_StarsE2) |
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491 done |
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492 |
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493 lemma PosOrd_Stars_append_eq: |
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494 assumes "flats vs1 = flats vs2" |
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495 shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2) \<longleftrightarrow> Stars vs1 :\<sqsubset>val Stars vs2" |
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496 using assms |
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497 apply(rule_tac iffI) |
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498 apply(erule PosOrd_Stars_appendE) |
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499 apply(rule PosOrd_Stars_appendI) |
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500 apply(auto) |
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501 done |
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502 |
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503 lemma PosOrd_almost_trichotomous: |
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504 shows "v1 :\<sqsubset>val v2 \<or> v2 :\<sqsubset>val v1 \<or> (length (flat v1) = length (flat v2))" |
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505 apply(auto simp add: PosOrd_ex_def) |
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506 apply(auto simp add: PosOrd_def) |
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507 apply(rule_tac x="[]" in exI) |
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508 apply(auto simp add: Pos_empty pflat_len_simps) |
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509 apply(drule_tac x="[]" in spec) |
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510 apply(auto simp add: Pos_empty pflat_len_simps) |
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511 done |
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512 |
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513 |
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514 |
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515 section \<open>The Posix Value is smaller than any other Value\<close> |
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516 |
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517 |
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518 lemma Posix_PosOrd: |
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519 assumes "s \<in> r \<rightarrow> v1" "v2 \<in> LV r s" |
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520 shows "v1 :\<sqsubseteq>val v2" |
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521 using assms |
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522 proof (induct arbitrary: v2 rule: Posix.induct) |
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523 case (Posix_ONE v) |
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524 have "v \<in> LV ONE []" by fact |
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525 then have "v = Void" |
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526 by (simp add: LV_simps) |
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527 then show "Void :\<sqsubseteq>val v" |
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528 by (simp add: PosOrd_ex_eq_def) |
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529 next |
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530 case (Posix_CH c v) |
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531 have "v \<in> LV (CH c) [c]" by fact |
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532 then have "v = Char c" |
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533 by (simp add: LV_simps) |
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534 then show "Char c :\<sqsubseteq>val v" |
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535 by (simp add: PosOrd_ex_eq_def) |
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536 next |
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537 case (Posix_ALT1 s r1 v r2 v2) |
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538 have as1: "s \<in> r1 \<rightarrow> v" by fact |
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539 have IH: "\<And>v2. v2 \<in> LV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact |
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540 have "v2 \<in> LV (ALT r1 r2) s" by fact |
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541 then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s" |
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542 by(auto simp add: LV_def prefix_list_def) |
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543 then consider |
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544 (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s" |
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545 | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s" |
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546 by (auto elim: Prf.cases) |
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547 then show "Left v :\<sqsubseteq>val v2" |
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548 proof(cases) |
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549 case (Left v3) |
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550 have "v3 \<in> LV r1 s" using Left(2,3) |
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551 by (auto simp add: LV_def prefix_list_def) |
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552 with IH have "v :\<sqsubseteq>val v3" by simp |
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553 moreover |
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554 have "flat v3 = flat v" using as1 Left(3) |
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555 by (simp add: Posix1(2)) |
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556 ultimately have "Left v :\<sqsubseteq>val Left v3" |
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557 by (simp add: PosOrd_ex_eq_def PosOrd_Left_eq) |
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558 then show "Left v :\<sqsubseteq>val v2" unfolding Left . |
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559 next |
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560 case (Right v3) |
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561 have "flat v3 = flat v" using as1 Right(3) |
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562 by (simp add: Posix1(2)) |
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563 then have "Left v :\<sqsubseteq>val Right v3" |
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564 unfolding PosOrd_ex_eq_def |
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565 by (simp add: PosOrd_Left_Right) |
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566 then show "Left v :\<sqsubseteq>val v2" unfolding Right . |
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567 qed |
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568 next |
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569 case (Posix_ALT2 s r2 v r1 v2) |
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570 have as1: "s \<in> r2 \<rightarrow> v" by fact |
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571 have as2: "s \<notin> L r1" by fact |
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572 have IH: "\<And>v2. v2 \<in> LV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact |
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573 have "v2 \<in> LV (ALT r1 r2) s" by fact |
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574 then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s" |
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575 by(auto simp add: LV_def prefix_list_def) |
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576 then consider |
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577 (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s" |
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578 | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s" |
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579 by (auto elim: Prf.cases) |
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580 then show "Right v :\<sqsubseteq>val v2" |
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581 proof (cases) |
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582 case (Right v3) |
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583 have "v3 \<in> LV r2 s" using Right(2,3) |
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584 by (auto simp add: LV_def prefix_list_def) |
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585 with IH have "v :\<sqsubseteq>val v3" by simp |
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586 moreover |
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587 have "flat v3 = flat v" using as1 Right(3) |
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588 by (simp add: Posix1(2)) |
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589 ultimately have "Right v :\<sqsubseteq>val Right v3" |
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590 by (auto simp add: PosOrd_ex_eq_def PosOrd_RightI) |
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591 then show "Right v :\<sqsubseteq>val v2" unfolding Right . |
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592 next |
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593 case (Left v3) |
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594 have "v3 \<in> LV r1 s" using Left(2,3) as2 |
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595 by (auto simp add: LV_def prefix_list_def) |
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596 then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3) |
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597 by (simp add: Posix1(2) LV_def) |
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598 then have "False" using as1 as2 Left |
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599 by (auto simp add: Posix1(2) L_flat_Prf1) |
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600 then show "Right v :\<sqsubseteq>val v2" by simp |
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601 qed |
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602 next |
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603 case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3) |
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604 have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+ |
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605 then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2)) |
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606 have IH1: "\<And>v3. v3 \<in> LV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact |
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607 have IH2: "\<And>v3. v3 \<in> LV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact |
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608 have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact |
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609 have "v3 \<in> LV (SEQ r1 r2) (s1 @ s2)" by fact |
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610 then obtain v3a v3b where eqs: |
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611 "v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2" |
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612 "flat v3a @ flat v3b = s1 @ s2" |
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613 by (force simp add: prefix_list_def LV_def elim: Prf.cases) |
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614 with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def |
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615 by (smt L_flat_Prf1 append_eq_append_conv2 append_self_conv) |
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616 then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs |
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617 by (simp add: sprefix_list_def append_eq_conv_conj) |
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618 then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)" |
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619 using PosOrd_spreI as1(1) eqs by blast |
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620 then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> LV r1 s1 \<and> v3b \<in> LV r2 s2)" using eqs(2,3) |
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621 by (auto simp add: LV_def) |
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622 then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast |
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623 then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1 |
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624 unfolding PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_Seq_eq) |
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625 then show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast |
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626 next |
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627 case (Posix_STAR1 s1 r v s2 vs v3) |
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628 have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+ |
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629 then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2)) |
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630 have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact |
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631 have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact |
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632 have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact |
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633 have cond2: "flat v \<noteq> []" by fact |
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634 have "v3 \<in> LV (STAR r) (s1 @ s2)" by fact |
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635 then consider |
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636 (NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)" |
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637 "\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r" |
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638 "flat (Stars (v3a # vs3)) = s1 @ s2" |
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639 | (Empty) "v3 = Stars []" |
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640 unfolding LV_def |
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641 apply(auto) |
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642 apply(erule Prf.cases) |
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643 apply(auto) |
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644 apply(case_tac vs) |
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645 apply(auto intro: Prf.intros) |
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646 done |
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647 then show "Stars (v # vs) :\<sqsubseteq>val v3" |
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648 proof (cases) |
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649 case (NonEmpty v3a vs3) |
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650 have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) . |
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651 with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3) |
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652 unfolding prefix_list_def |
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653 by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7)) |
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654 then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4) |
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655 by (simp add: sprefix_list_def append_eq_conv_conj) |
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656 then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" |
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657 using PosOrd_spreI as1(1) NonEmpty(4) by blast |
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658 then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)" |
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659 using NonEmpty(2,3) by (auto simp add: LV_def) |
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660 then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast |
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661 then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" |
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662 unfolding PosOrd_ex_eq_def by auto |
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663 then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1 |
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664 unfolding PosOrd_ex_eq_def |
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665 using PosOrd_StarsI PosOrd_StarsI2 by auto |
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666 then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast |
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667 next |
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668 case Empty |
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669 have "v3 = Stars []" by fact |
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670 then show "Stars (v # vs) :\<sqsubseteq>val v3" |
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671 unfolding PosOrd_ex_eq_def using cond2 |
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672 by (simp add: PosOrd_shorterI) |
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673 qed |
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674 next |
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675 case (Posix_STAR2 r v2) |
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676 have "v2 \<in> LV (STAR r) []" by fact |
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677 then have "v2 = Stars []" |
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678 unfolding LV_def by (auto elim: Prf.cases) |
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679 then show "Stars [] :\<sqsubseteq>val v2" |
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680 by (simp add: PosOrd_ex_eq_def) |
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681 qed |
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682 |
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683 |
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684 lemma Posix_PosOrd_reverse: |
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685 assumes "s \<in> r \<rightarrow> v1" |
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686 shows "\<not>(\<exists>v2 \<in> LV r s. v2 :\<sqsubset>val v1)" |
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687 using assms |
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688 by (metis Posix_PosOrd less_irrefl PosOrd_def |
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689 PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans) |
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690 |
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691 lemma PosOrd_Posix: |
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692 assumes "v1 \<in> LV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1" |
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693 shows "s \<in> r \<rightarrow> v1" |
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694 proof - |
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695 have "s \<in> L r" using assms(1) unfolding LV_def |
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696 using L_flat_Prf1 by blast |
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697 then obtain vposix where vp: "s \<in> r \<rightarrow> vposix" |
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698 using lexer_correct_Some by blast |
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699 with assms(1) have "vposix :\<sqsubseteq>val v1" by (simp add: Posix_PosOrd) |
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700 then have "vposix = v1 \<or> vposix :\<sqsubset>val v1" unfolding PosOrd_ex_eq2 by auto |
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701 moreover |
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702 { assume "vposix :\<sqsubset>val v1" |
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703 moreover |
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704 have "vposix \<in> LV r s" using vp |
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705 using Posix_LV by blast |
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706 ultimately have "False" using assms(2) by blast |
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707 } |
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708 ultimately show "s \<in> r \<rightarrow> v1" using vp by blast |
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709 qed |
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710 |
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711 lemma Least_existence: |
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712 assumes "LV r s \<noteq> {}" |
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713 shows " \<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v" |
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714 proof - |
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715 from assms |
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716 obtain vposix where "s \<in> r \<rightarrow> vposix" |
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717 unfolding LV_def |
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718 using L_flat_Prf1 lexer_correct_Some by blast |
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719 then have "\<forall>v \<in> LV r s. vposix :\<sqsubseteq>val v" |
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720 by (simp add: Posix_PosOrd) |
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721 then show "\<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v" |
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722 using Posix_LV \<open>s \<in> r \<rightarrow> vposix\<close> by blast |
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723 qed |
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724 |
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725 lemma Least_existence1: |
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726 assumes "LV r s \<noteq> {}" |
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727 shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v" |
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728 using Least_existence[OF assms] assms |
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729 using PosOrdeq_antisym by blast |
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730 |
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731 lemma Least_existence2: |
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732 assumes "LV r s \<noteq> {}" |
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733 shows " \<exists>!vmin \<in> LV r s. lexer r s = Some vmin \<and> (\<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v)" |
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734 using Least_existence[OF assms] assms |
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735 using PosOrdeq_antisym |
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736 using PosOrd_Posix PosOrd_ex_eq2 lexer_correctness(1) by auto |
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737 |
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738 |
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739 lemma Least_existence1_pre: |
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740 assumes "LV r s \<noteq> {}" |
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741 shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> (LV r s \<union> {v'. flat v' \<sqsubset>spre s}). vmin :\<sqsubseteq>val v" |
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742 using Least_existence[OF assms] assms |
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743 apply - |
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744 apply(erule bexE) |
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745 apply(rule_tac a="vmin" in ex1I) |
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746 apply(auto)[1] |
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747 apply (metis PosOrd_Posix PosOrd_ex_eq2 PosOrd_spreI PosOrdeq_antisym Posix1(2)) |
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748 apply(auto)[1] |
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749 apply(simp add: PosOrdeq_antisym) |
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750 done |
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751 |
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752 lemma |
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753 shows "partial_order_on UNIV {(v1, v2). v1 :\<sqsubseteq>val v2}" |
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754 apply(simp add: partial_order_on_def) |
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755 apply(simp add: preorder_on_def refl_on_def) |
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756 apply(simp add: PosOrdeq_refl) |
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757 apply(auto) |
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758 apply(rule transI) |
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759 apply(auto intro: PosOrdeq_trans)[1] |
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760 apply(rule antisymI) |
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761 apply(simp add: PosOrdeq_antisym) |
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762 done |
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763 |
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764 lemma |
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765 "wf {(v1, v2). v1 :\<sqsubset>val v2 \<and> v1 \<in> LV r s \<and> v2 \<in> LV r s}" |
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766 apply(rule finite_acyclic_wf) |
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767 prefer 2 |
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768 apply(simp add: acyclic_def) |
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769 apply(induct_tac rule: trancl.induct) |
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770 apply(auto)[1] |
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771 oops |
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772 |
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773 |
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774 unused_thms |
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775 |
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776 end |